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Metamath Proof Explorer

Theorem aspss

Description: Span preserves subset ordering. ( spanss analog.) (Contributed by Mario Carneiro, 7-Jan-2015)

Ref Expression
Hypotheses aspval.a ⊢ 𝐴 = ( AlgSpan ‘ 𝑊 )
aspval.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
Assertion aspss ( ( 𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑆 ) → ( 𝐴 ‘ 𝑇 ) ⊆ ( 𝐴 ‘ 𝑆 ) )

Proof

Step Hyp Ref Expression
1 aspval.a ⊢ 𝐴 = ( AlgSpan ‘ 𝑊 )
2 aspval.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
3 simpl3 ⊢ ( ( ( 𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑆 ) ∧ 𝑡 ∈ ( ( SubRing ‘ 𝑊 ) ∩ ( LSubSp ‘ 𝑊 ) ) ) → 𝑇 ⊆ 𝑆 )
4 sstr2 ⊢ ( 𝑇 ⊆ 𝑆 → ( 𝑆 ⊆ 𝑡 → 𝑇 ⊆ 𝑡 ) )
5 3 4 syl ⊢ ( ( ( 𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑆 ) ∧ 𝑡 ∈ ( ( SubRing ‘ 𝑊 ) ∩ ( LSubSp ‘ 𝑊 ) ) ) → ( 𝑆 ⊆ 𝑡 → 𝑇 ⊆ 𝑡 ) )
6 5 ss2rabdv ⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑆 ) → { 𝑡 ∈ ( ( SubRing ‘ 𝑊 ) ∩ ( LSubSp ‘ 𝑊 ) ) ∣ 𝑆 ⊆ 𝑡 } ⊆ { 𝑡 ∈ ( ( SubRing ‘ 𝑊 ) ∩ ( LSubSp ‘ 𝑊 ) ) ∣ 𝑇 ⊆ 𝑡 } )
7 intss ⊢ ( { 𝑡 ∈ ( ( SubRing ‘ 𝑊 ) ∩ ( LSubSp ‘ 𝑊 ) ) ∣ 𝑆 ⊆ 𝑡 } ⊆ { 𝑡 ∈ ( ( SubRing ‘ 𝑊 ) ∩ ( LSubSp ‘ 𝑊 ) ) ∣ 𝑇 ⊆ 𝑡 } → ∩ { 𝑡 ∈ ( ( SubRing ‘ 𝑊 ) ∩ ( LSubSp ‘ 𝑊 ) ) ∣ 𝑇 ⊆ 𝑡 } ⊆ ∩ { 𝑡 ∈ ( ( SubRing ‘ 𝑊 ) ∩ ( LSubSp ‘ 𝑊 ) ) ∣ 𝑆 ⊆ 𝑡 } )
8 6 7 syl ⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑆 ) → ∩ { 𝑡 ∈ ( ( SubRing ‘ 𝑊 ) ∩ ( LSubSp ‘ 𝑊 ) ) ∣ 𝑇 ⊆ 𝑡 } ⊆ ∩ { 𝑡 ∈ ( ( SubRing ‘ 𝑊 ) ∩ ( LSubSp ‘ 𝑊 ) ) ∣ 𝑆 ⊆ 𝑡 } )
9 simp1 ⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑆 ) → 𝑊 ∈ AssAlg )
10 simp3 ⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑆 ) → 𝑇 ⊆ 𝑆 )
11 simp2 ⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑆 ) → 𝑆 ⊆ 𝑉 )
12 10 11 sstrd ⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑆 ) → 𝑇 ⊆ 𝑉 )
13 eqid ⊢ ( LSubSp ‘ 𝑊 ) = ( LSubSp ‘ 𝑊 )
14 1 2 13 aspval ⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑇 ⊆ 𝑉 ) → ( 𝐴 ‘ 𝑇 ) = ∩ { 𝑡 ∈ ( ( SubRing ‘ 𝑊 ) ∩ ( LSubSp ‘ 𝑊 ) ) ∣ 𝑇 ⊆ 𝑡 } )
15 9 12 14 syl2anc ⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑆 ) → ( 𝐴 ‘ 𝑇 ) = ∩ { 𝑡 ∈ ( ( SubRing ‘ 𝑊 ) ∩ ( LSubSp ‘ 𝑊 ) ) ∣ 𝑇 ⊆ 𝑡 } )
16 1 2 13 aspval ⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉 ) → ( 𝐴 ‘ 𝑆 ) = ∩ { 𝑡 ∈ ( ( SubRing ‘ 𝑊 ) ∩ ( LSubSp ‘ 𝑊 ) ) ∣ 𝑆 ⊆ 𝑡 } )
17 16 3adant3 ⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑆 ) → ( 𝐴 ‘ 𝑆 ) = ∩ { 𝑡 ∈ ( ( SubRing ‘ 𝑊 ) ∩ ( LSubSp ‘ 𝑊 ) ) ∣ 𝑆 ⊆ 𝑡 } )
18 8 15 17 3sstr4d ⊢ ( ( 𝑊 ∈ AssAlg ∧ 𝑆 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑆 ) → ( 𝐴 ‘ 𝑇 ) ⊆ ( 𝐴 ‘ 𝑆 ) )